Constructing abelian varieties from rank 2 Galois representations

Let U be a smooth affine curve over a number field K with a compactification X and let L be a rank 2, geometrically irreducible lisse Ql-sheaf on U with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field E ⊂ Ql, and has bad, infinite reduction at some closed point x of X\U. We show that L occurs as a summand of the cohomology of a family of abelian varieties over U. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when E=Q, then L is isomorphic to the cohomology of an elliptic curve EU -> U.
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